with \(\Theta _\infty \)  the applied field infinitely far from the particle. Phoretic motion has several key characteristics. First the motion takes its origin within the interfacial diffuse layer close to the particle: typically the electric double layer for charged particles, but any other surface interaction characterized by a diffuse interface of finite thickness. Within this layer the fluid is displaced relatively to the particle due e.g. to electro-osmotic or diffusio-osmotic transport; see Fig. REF  for an illustration  [1]}, [2]}. Second, motion of the particle is force-free, i.e. the global force on the particle is zero, the particle moves at a steady velocity.
This can be understood in simple terms for example for electrophoresis: the cloud of counter-ions around the particle experiences a force due to the electric field which is opposite to that applied directly to the particle, so that the total force acting on the system of the particle and its ionic diffuse layer experiences a vanishing total
force.
Both electro- and diffusio- phoresis and correspondingly electro- and diffusio-osmosis can all be interpreted as a single osmotic phenomena, since the two are related via a unique driving field, the electro-chemical potential [3]}.
